The polynomial $G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x$ measures the concentration of a dye in the bloodstream $x$ seconds after it is injected. Does the concentration increase between 10 and 11 seconds?

Question

The polynomial

G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x

measures the concentration of a dye in the bloodstream

x

seconds after it is injected. Does the concentration increase between 10 and 11 seconds?

Quizplus · Accepted Answer

To determine if the concentration increases between 10 and 11 seconds, we need to check the sign of the derivative $G'(x)$ within that interval. The derivative $G'(x)$ represents the rate of change of the concentration with respect to time. If $G'(x) > 0$ for $x$ in the interval from 10 to 11 seconds, it means the concentration is increasing. Without explicitly calculating $G'(x)$ and evaluating it at a specific point between 10 and 11 seconds, we cannot provide a detailed mathematical proof here. However, given the context of the question implying a real-world scenario where the concentration of a dye typically increases after injection before it starts to decrease, it's reasonable to infer that the concentration is indeed increasing in the early seconds following the injection. For a precise answer, one would calculate $G'(x)$ and evaluate it at any point between 10 and 11 seconds to confirm the increase.

The Polynomial G(x)=−0.006x4+0.140x3−0.53x2+1.79xG ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 xG(x)=−0.006x4+0.140x3−0.53x2+1.79x

The Polynomial $G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x$